In a communication system with a multi-antenna transmitter and, for example, a single-antenna receiver, the transmitter may have an estimate of the channel between each of its antennas and the receiver antenna. Alternatively, in case of multi antenna receiver the transmitter can assume a receiver beamforming vector that emulates a single effective antenna. To optimize the signal-to-noise ratio (SNR) at the receiver, the transmitter should appropriately design the TX beamforming vector, that is, the vector of complex coefficients that multiply the single data symbol before sending it for transmission in each antenna.
The designed beamforming vector is subject to physical and regulatory constraints. The physical constraints may include a limitation on the TX power of each power amplifier (PA), a limitation on the overall TX power due to packaging and thermal constraints, etc. The regulatory constraints include any limitation imposed by a regulator, such as FCC or ETSI. Typically, regulatory constraints include limitations on the overall line of site (LOS) effective isotropic radiated power (EIRP) and the overall TX power.
Thus, practical beamforming design problem is typically subject to three types of constraints: per radio-frequency (RF) chain constraints, also referred to as Per-antenna power constraints, an overall power constraint, and an EIRP constraint. Therefore, it is desirable to have an efficient method to design a beamforming vector that satisfies all constraints while substantially maximizing SNR.
Additionally practical beamforming design problem may be subject to a subset of two constraints. For example, ETSI standard enforces only EIRP constraint. However, per-antenna power constraint may arise from physical limitation of the practical communication system. Thus, a practical communication system that conforms to ETSI standard may require a beamforming design under per-antenna power constraints and EIRP constraint. Moreover, in many practical situations, satisfying one constraint does not guarantee conforming to the other constraint. For example, in the 5470-5725 MHz band, the EIRP limitation of ETSI is 1 W. For a system with 4 isotropic antennas and with 100 mW power amplifiers, satisfying only the per-antenna constraints does not guarantee satisfying the EIRP constraint, because if all 4 antennas transmit in full power, the EIRP might be as high as 42·0.1=1.6 W. In addition, satisfying only the EIRP constraint does not guarantee satisfying the per-antenna constraints, as transmitting from a single antenna in 1 W satisfies the EIRP constraint but not the per-antenna constraints. Thus, a beamforming design under the per-antenna power constraints and an EIRP constraint is needed.
According to the following scenario, beamforming design problem may be subject to per-antenna and the overall power constraints. In FCC both the EIRP and the overall power are limited, and the limit on the EIRP is 6 dB higher than the limit on the total power. For a system with 3 isotropic antennas, satisfying the total power constraint assures that the EIRP constraint is satisfied. Hence, it follows that the only effective regulatory constraint is the total power constraint. Writing P for the total power constraint, if the maximum allowed per-antenna power levels are in the interval (P/3, P), both the per-antenna and the overall power constraints should be considered. Thus, a beamforming design under per-antenna and overall power constraints is needed.
The following example presents a case where all three types of constraints should be accounted for. Consider transmission in the 5.25 GHz-5.35 GHz sub-band under FCC regulations. In this sub-band, FCC requires that total TX power is below 24 dBm, and that EIRP is below 30 dBm. Suppose that the maximum output power of each power amplifier of the transmitter is 22 dBm, and that the transmitter has 4 antennas, each with a gain of 1.5 dBi. Here, satisfying two constraints does not assure that the third constraint is also satisfied. Hence a beamforming design under all three constraint types is needed.
Unfortunately, existing methods are either far from optimum or too complicated for real time software implementation. On the one side of the scale, there are methods that start with well-known solutions to a single type of constraint, and then scale down the entire beamforming vector to meet the other constraints. On the other side of the scale, there are convex optimization methods that are very complicated both computationally and conceptually.
In detail, when the only limitation is on the total power, it is well known in the art that the optimum transmit beamforming vector is the maximum ratio combining (MRC) vector related to the channel vector and the maximum allowed power. Also, if only the per-antenna powers are constrained, then an optimum beamforming vector is obtained by using all available power of each antenna, while choosing phases that assure coherent addition at the receiver. Such a beamforming vector will be referred to as a full per-antenna power (FPAP) vector throughout this application.
In a communication system with multi antenna receivers the equivalent of an MRC vector is typically computed using the singular value decomposition (SVD) of the channel matrix. The first column of the transmit matrix V, obtained from the SVD, corresponding to the largest singular value of the channel, is the transmit MRC vector to the best RX effective antenna. According to IEEE 802.11n/ac explicit matrix feedback, the receiver usually returns the V matrix to the transmitter.
A simple, yet suboptimum, method for finding the beamforming vector in case of multiple constraints is to start by considering only the total power or the per-antenna power constraints, and then to scale the resulting MRC or FPAP vector in order to satisfy the other constraints. While simple, this method, referred to as the scaling method throughout this application, is typically quite far from optimal.
Alternatively, it is possible to aim at the true optimum of the beamforming design problem, at the cost of considerably higher complexity. Since this problem is a convex optimization problem, existing convex optimizations algorithms can be used to solve it. While such algorithms have time complexity that is polynomial in the number of variables, they are still considerably more complicated than the above MRC/FPAP+ scaling solutions.